Summary: On Pseudorandom Generators with
Linear Stretch in NC 0 #
Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz
Computer Science Department, Technion, Haifa 32000, Israel
{abenny,yuvali,eyalk}@cs.technion.ac.il
Abstract. We consider the question of constructing cryptographic pseudoran­
dom generators (PRGs) in NC 0 , namely ones in which each bit of the output
depends on just a constant number of input bits. Previous constructions of such
PRGs were limited to stretching a seed of n bits to n + o(n) bits. This leaves
open the existence of a PRG with a linear (let alone superlinear) stretch in NC 0 .
In this work we study this question and obtain the following main results:
1. We show that the existence of a linear­stretch PRG in NC 0 implies non­
trivial hardness of approximation results without relying on PCP machinery.
In particular, that Max 3SAT is hard to approximate to within some constant.
2. We construct a linear­stretch PRG in NC 0 under a specific intractability as­
sumption related to the hardness of decoding ``sparsely generated'' linear
codes. Such an assumption was previously conjectured by Alekhnovich [1].
We note that Alekhnovich directly obtains hardness of approximation results from
the latter assumption. Thus, we do not prove hardness of approximation under
new concrete assumptions. However, our first result is motivated by the hope to